Parabolic PDE¶

Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner

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Heat equation:
$$u_t - c u_{xx} = 0\quad \text{in}\, \Omega\, \times\, [0,T]$$ $$u(0,x) = g(x)\quad \text{in}\, \Omega\,$$ $$u(t,0) = u(t,1) = 0$$ Let: $$g(x) = sin(\pi x)$$

In [2]:
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import math
import sympy as sym


Solution¶

$$u(t,x) = e^{-\pi^2t}sin(\pi x)$$

Check symbolically.

In [3]:
sx = sym.Symbol("x")
st = sym.Symbol("t")
su = sym.exp(-sym.pi**2*st) * sym.sin(sym.pi * sx)

sym.diff(su, st) - sym.diff(su, sx, 2)

Out[3]:
$\displaystyle 0$

Set Up the Grid¶

• dx: grid spacing in the $x$-direction
• x: grid coordinates
In [4]:
nx = 200
x = np.linspace(0, 1, nx, endpoint=False)
dx = x[1] - x[0]


Set Time Parameters¶

In [5]:
T = 1.0 # end time
dt = 0.009
nt = int(T/dt)
print('T = %g' % T)
print('tsteps = %d' % nt)
print('    dx = %g' % dx)
print('    dt = %g' % dt)

T = 1
tsteps = 111
dx = 0.005
dt = 0.009


Set Initial Condition¶

In [6]:
PI = math.pi
def g(x):
return np.sin(PI*x)


Plot Solution¶

In [7]:
import time

plotit = True
u = g(x)

fig = plt.figure(figsize=(8,8))
plt.title('u vs x')
line1, = plt.plot(x, u, lw=3, clip_on=False)

def timestepper(n):
uex = np.exp(-1*PI**2*(n+1)*dt)*g(x)
line1.set_data(x, uex)
return line1

from matplotlib import animation
from IPython.display import HTML

ani = animation.FuncAnimation(fig, timestepper, frames=nt, interval=20)
html = HTML(ani.to_jshtml())
plt.clf()
html

Out[7]: